F(s, Sob) = v(s)v(sob)fvv in .NET

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6 Correlation Function of Fields
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[2"Y,Bp(s, Sob) - "Y,Bv(S, Sob) - "Y,Bh(S, Sob)] ,B=v,h = A 41f 0 {2Re(Jvvf:h)
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sin Os (3.5.75a)
[(Ifvv(s, sob)1
+ (Ifhv(s, sOb)1 2 )] }
TBh(So) =T{l- A 1 0 [27r d<ps [7r/2 dOssinO s
. [(lfvh(S, Sob) 1
UB(So) = T {A
+ (lfhh(S, sOb)1 2 )]}
c~s 0
127r d<ps
7r 2 / dOs sin Os [2Re(fvv(s, Sob)!:h (s, Sob))
+ 2Re(fhv(s, Sob)!hh(S, SOb))]}
VB (so)
T{ A
c~s 0
/2 dOs sin Os [2Im(Jvv (s, Sob) f: h(s, Sob))
+ 2Im(fhv(s, Sob)!hh(S, SOb))]}
6 Correlation Function of Fields
Because random medium statistics are described by correlation functions, it will be useful to study the correlation of electromagnetic fields. The correlation of electromagnetic fields can often be related to the correlation properties of the random medium or the random rough surface.
We have previously studied the third and fourth Stokes parameters, which represent the correlation between the vertical and horizontal polarizations. In this section, we discuss other useful field correlation functions for random scattering.
Angular Correlation Function
The angular correlation function (ACF) describes the correlation of scattered fields in two different directions. Consider a scattered field in direction ks1 with an incident wave in direction kilo The scattered field is denoted by 'l/J(k sl , kil)' Consider another scattered field 'l/J(k s2 , ki2 ) in direction ks2 due to incident wave in direction ki2. Then the angular correlation function is described by
It will be shown in later chapters that if the medium is statistically translational invariant in the horizontal direction - for example, with random permittivity E(X, y, z) such that
(E(XI, YI, ZI)E(X2, Y2, Z2))
= 8(XI
- x2)8(YI - Y2)R(XI, YI; Zl, Z2)
where R is some correlation function of the random medium ACF is zero unless
then the
This can be regarded as a phase matching condition for the angular correlation function for statistical inhomogeneous medium. It is also called the "memory effect."
Mutual Coherent Function
Consider an incident wave at a carrier frequency
Wo .
In general the field
u(r, t) is u(r, t) = Re['l/J(r, t)e- iwot ]
where 'l/J(r, t) is the complex field.
'l/J = 'l/Jc + 'l/Jf when 'l/Jc = ('l/J)
The correlation function time points:
is the correlation of the fields at two space and
Correlation Function of Fields
It can be decomposed into a coherent part and an incoherent part.
f = f c + fi fcCfl, tl;r2, t2) = ('l/J(TI, tl)) ('l/J*(T2, t2)) fi(rl, tl;r2, t2) = ('l/J/(rl, t l )) ('l/Jj(r2, t2)) (3.6.7)
(3.6.8) (3.6.9)
Consider the case that TI = r2 = T and the random medium is stationary in time. Then f i depends only on the time difference
f - f (r , T)
where T = tl - t2 is the time difference. The temporal frequency spectrum is the Fourier transform of the correlation function
Wi(r, w) =
dTfi(r, T)e
Two Frequency Mutual Coherent Function
The concept of correlation function can be generalized to two frequencies. Let u(w, T, t) be the field where the incident wave is a time harmonic function of e-iwt . Then u(w, T, t) = Re('lj!(w, T, t)e- iwt ) (3.6.13) The two-frequency mutual coherent function is
r(WI,TI,tl;w2,T2,t2) = ('lj!(WI,TI,tI)'lj!*(W2,T2,t2))
It can also be decomposed into a coherent part and an incoherent part
r c = ('lj!(WI, TI, tl)) ('lj!*(W2, T2, t2))
f=fc+f i
Relation Between Correlation Function and Specific Intensity
We can write the fluctuating field 'l/J/(r) as consisting of a spectrum of waves going in all 41f directions
'l/J/(rl) = 'lj!j(T2)
d'S;j; j(r2, s)e-iks.r2
r /('r1, r2)
= ('l/J/(r1)'l/Jj(r2))
dS l
ds 2({;/(r1, Sl){;j(r2, 82))eiksl'Tle-iks2'T2
For fluctuating fields, it is usually the case that
({J/("r1, 81){Jj(r2' 82)) = I(r a, 81)02(81 - 82)
where I is the specific intensity, 02 is the two-dimensional angular Dirac delta function, and r a = (r1 + r2)/2. Equation (3.6.20) means that the correlation function in spectral domain is uncorrelated for two different directions and the correlation is also zero if r1 and r2 are far apart. Putting (3.6.20) in (3.6.19) gives
r!Cr1,r2) =
showing that the field correlation function is an angular transform of the specific intensity.
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